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MAT 153: College Algebra and Trigonometry

Section 2.9 More Exponents and Logarithms

Goals:
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  • E: Be able to solve an equation with an unknown exponent.
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  • E: Be able to model a situation with appropriate exponential equation(s) and interpret the solution.
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  • E: Be able to determine the equation of an exponential function given a table of values.
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  • E: Be able to use definition and properties of logarithms to rewrite expressions involving logarithms in different forms.
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  • F: Be able to determine the inverse of a function given in any form (graph, table, equation).
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Problem 2.9.1.

Use the definition of logarithm to solve the following equations.
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  1. \(\displaystyle 4 \cdot 7^{3x} = 9604\)
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  2. \(\displaystyle 10 - 3^{5x} = -719\)
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  3. \(\displaystyle 5^{2x} - 5^x - 30 = 0\)
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Problem 2.9.2.

A bank is offering to pay \(2.5\%\) interest compounded annually. Clare opens an account with $1200.
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  1. What will be Clare’s balance in 5 years?
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  2. Write a function equation, \(C(t)\text{,}\) for Clare’s balance after \(t\) years.
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  3. When will Clare’s balance reach $1500? First give an exact answer, then give a decimal approximation.
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Problem 2.9.3.

Rewrite each expression using a single natural logarithm.
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  1. \(\displaystyle \ln (x-3) - 4\ln (x+1)\)
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  2. \(\displaystyle \ln (x^3y) + 3\ln (z) - 2\ln (y)\)
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Problem 2.9.4.

An epidemic is sweeping through a population of rats, so that the number of rats infected is doubling every three days. When scientists first observed the infection, they estimated that there were 120 rats infected.
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  1. Write a function equation \(I(t)\) for the number of rats infected \(t\) days after the initial observation.
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  2. Rewrite \(I(t)\) using the natural base, \(e\text{.}\)
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  3. How many rats will be infected 30 days after the initial observation, assuming the infection continues to spread at the same rate?
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  4. When will the number of infected rats reach 200,000? First give an exact answer, and then give a decimal approximation.
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Problem 2.9.5.

Refer to the population of Shanghai, China, for given years as shown in Table 2.9.6.
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Table 2.9.6. Population, \(P(t)\text{,}\) of Shanghai \(t\) years after 1990
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\(t\) 0 10 20 22
\(P(t)\) 13,341,900 16,407,700 23,019,200 23,710,000
  1. Using the population of Shanghai in 1990 and 2012, build an exponential model for \(P(t)\text{.}\)
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  2. Use your exponential model to predict the population of Shanghai in 2000. How does your prediction compare with the actual population at that time?
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  3. Use your exponential model to predict in what year the population of Shanghai will be 25,000,000. First give your answer in exact form using a logarithm, and then give the decimal approximation.
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Problem 2.9.7.

The amount of money accumulated in a bank account where the interest is compounded continually after \(t\) years at interest rate \(r\) with an initial amount \(P\) is given by the function
\begin{equation*} A(t)=Pe^{rt} \end{equation*}
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  1. If you put $800 in a bank account with interest rate 5% that is compounded continually, write a function equation for \(A(t)\text{.}\)
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  2. How much money will be in the account after 5 years?
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  3. How long will it take the account to reach $2,000?
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  4. Find an equation for the inverse function that gives the time, \(t\) as a function of the amount of money in the account, \(A\text{.}\)
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